I need help on this question, I know the answer (or, at least I think I know it), but I can't think of the rest.
Bobby claims that only upright squares can have rational numbers for side lengths. Is he correct? Explain.
I got yes, he is correct, but I can't hink of an explanation besides common sense! Help, please?
What is an upright square?
On a grid, an upright square is a square that is well, upright, on one of its sides, with the sides going along the grid points.
To determine whether Bobby's claim is correct, we need to understand what it means for a square to be "upright" and why it would have rational side lengths.
An upright square is a square that has its sides aligned vertically and horizontally, forming a right angle with the ground.
On the other hand, rational numbers are the numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero.
To explain whether Bobby is correct, we need to consider all possible squares. We can do this by considering both upright squares and non-upright squares.
Let's start with upright squares. If we consider a square with side length "a," its perpendicular sides will have the same length "a." In such a case, both the length and width are equal, resulting in the same side length for any upright square. Since we are dealing with equivalent lengths, it is plausible to conclude that the side length of an upright square will be a rational number. This is because any whole number can be expressed as a fraction with a denominator of 1.
Now, let's consider non-upright squares. These squares have side lengths arranged at an angle that is not vertical or horizontal. In such cases, the length and width of the square will be different, and it is challenging to determine a specific relationship between the lengths. Non-upright squares can have side lengths that are irrational numbers, meaning they cannot be expressed as fractions. For example, the diagonal length of a square with side length 1 is an irrational number (√2). Therefore, non-upright squares can have irrational side lengths.
In conclusion, Bobby's claim that only upright squares can have rational side lengths is incorrect. Non-upright squares can have irrational side lengths, while upright squares will indeed have rational side lengths. So, it is not solely based on common sense but requires a logical understanding of square geometry and number classification.